From fluttering to drifting in inertialess sedimentation of achiral particles

1. Problem Statement

The paper investigates the sedimentation dynamics of achiral particles (particles lacking chirality) in a viscous fluid at low Reynolds numbers (Stokes flow), where inertial effects are negligible.

• Context: While chiral particles (e.g., helices) naturally follow chiral trajectories, certain achiral particles with specific symmetries can also exhibit chiral motion due to translation-rotation coupling.
• Specific Focus: The study focuses on circular disks deformed into a U-shape. Previous work established that doubly-symmetric U-shaped disks (formed by bending a disk into a cylinder) act as "flutterers," sedimenting along complex, quasi-periodic spiral trajectories determined by their initial orientation.
• Research Question: How does the sedimentation behavior change when the symmetry of the U-shaped disk is broken? Specifically, what happens when the disk is "pinched" (deformed into a cone rather than a cylinder), reducing its symmetry from two planes to one?

2. Methodology

The authors employed a combined approach of numerical simulations and physical experiments.

• Mathematical Framework:
  • The motion is governed by the Resistance Matrix formalism, which links the particle's translational and rotational velocities to the external forces (gravity/buoyancy) and torques.
  • The system is reduced to a set of six autonomous Ordinary Differential Equations (ODEs) describing the position of the center of mass ($r_M$) and the orientation (defined by Tait-Bryan angles: yaw $\chi_{yaw}$, pitch $\chi_{pitch}$, and roll $\chi_{roll}$).
  • Crucially, the evolution of pitch and roll angles decouples from yaw and position, allowing the system's stability to be analyzed in a 2D phase plane $(\chi_{pitch}, \chi_{roll})$.
• Numerical Simulations:
  • The resistance matrix coefficients were computed using an augmented finite-element method (using the oomph-lib library) to solve 3D Stokes equations, specifically resolving singularities at the disk edges.
  • Simulations tracked the evolution of "pinched" disks where the minimum radius of curvature ($R_{min}$) was reduced while the maximum ($R_{max}$) was held constant, effectively increasing the degree of pinching.
• Experiments:
  • Physical experiments were conducted using polyamide nylon disks (thickness ~237 $\mu$m) fabricated with specific pinched geometries.
  • Disks were sedimented in a large cuboidal tank filled with a viscous fluid.
  • Particle orientation and trajectory were tracked over vertical distances of ~30 radii ($R$), ensuring the Reynolds number remained low ($Re \sim 10^{-2}$).

3. Key Contributions

1. Symmetry Breaking Analysis: The paper demonstrates that breaking the fore-aft symmetry of a U-shaped disk (via pinching) fundamentally alters the stability of its sedimentation trajectories.
2. Bifurcation Identification: The authors identify a pitchfork bifurcation that governs the transition between different sedimentation regimes. This occurs as the pinching parameter ($R_{min}$) changes, causing fixed points in the phase plane to coalesce and change stability.
3. Universal Behavior: They establish that for a wide range of pinched shapes, the sedimentation behavior converges to a globally attractive helical trajectory, regardless of the initial orientation.
4. Transition to "Drifters": The study defines the critical threshold where the particle transitions from a "flutterer" (spiraling) to a "drifter" (steady, non-rotating descent).

4. Key Results

A. Phase Plane Dynamics

• Doubly-Symmetric Case ($R_{min} = R_{max}$): The phase plane contains neutrally stable centers. Disks follow closed loops, alternating between pitching and rolling, resulting in complex, quasi-periodic spiral paths ("fluttering").
• Weakly Pinched Case ($R_{min} < R_{max}$ but close): The centers transform into stable spiral nodes. The disk's inclination evolves towards a fixed point $(\chi^*_{pitch}, \chi^*_{roll})$. The trajectory becomes a helix with a constant radius and pitch. The disk settles in a "nose-down" orientation while rotating about the gravity vector.
• Strongly Pinched Case: As pinching increases, the stable spiral nodes transition into stable nodes (monotonic approach). The radius of the helical trajectory increases.
• Critical Bifurcation Point: At a specific pinching threshold, the stable nodes coalesce with saddle points on the $\chi_{roll}=0$ axis.
  • At this point, $\chi^*_{roll} \to 0$, implying $d\chi_{yaw}/dt = 0$.
  • The disk ceases to rotate and settles along a straight trajectory with constant velocity. The particle has transitioned from a "flutterer" to a "drifter."

B. Experimental Validation

• Experimental data for pinched disks ($R_{min}=0.32, R_{max}=0.6$) matched the numerical predictions.
• The experiments confirmed that the disk settles into a stable "nose-down" helical trajectory regardless of its initial release orientation.
• The observed phase plane paths (extrapolated from short experimental segments) aligned with the theoretical fixed points (stable nodes and saddle points).

C. Transition Criteria

The transition from helical to straight trajectories is driven by the radius of the helix becoming infinite as the roll angle of the fixed point approaches zero. This is a direct consequence of the symmetry breaking induced by the pinching geometry.

5. Significance and Implications

• Robustness of Chiral Motion: The study confirms that chiral sedimentation (helical paths) is a robust phenomenon for achiral particles with broken symmetry, persisting over a wide range of geometries.
• Control of Suspension Dispersion: The findings have significant implications for the dispersion of dilute suspensions:
  • Flutterers/Helical Drifters: Sediment with zero net horizontal dispersion because their spiral paths cancel out lateral movement over time.
  • Drifters: Sediment along straight, inclined lines, leading to significant horizontal spreading (dispersion) as they settle.
• Applications: This understanding is crucial for applications involving:
  • Separation of biological molecules and 2D materials (e.g., graphene flakes).
  • Design of micro-swimmers.
  • Removal of microplastics and colloidal suspensions.
  • Blood testing and filtration processes.

In conclusion, the paper provides a comprehensive theoretical and experimental framework for predicting how geometric symmetry breaking dictates the transition from complex, oscillatory sedimentation to steady, drifting motion in low-Reynolds-number flows.